Solve for $x$ : $ 7|x + 3| - 7 = 6|x + 3| + 2 $
Answer: Subtract $ {6|x + 3|} $ from both sides: $ \begin{eqnarray} 7|x + 3| - 7 &=& 6|x + 3| + 2 \\ \\ { - 6|x + 3|} && { - 6|x + 3|} \\ \\ 1|x + 3| - 7 &=& 2 \end{eqnarray} $ Add ${7}$ to both sides: $ \begin{eqnarray} 1|x + 3| - 7 &=& 2 \\ \\ { + 7} &=& { + 7} \\ \\ 1|x + 3| &=& 9 \end{eqnarray} $ Simplify: $ |x + 3| = 9$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 3 = -9 $ or $ x + 3 = 9 $ Solve for the solution where $x + 3$ is negative: $ x + 3 = -9 $ Subtract ${3}$ from both sides: $ \begin{eqnarray} x + 3 &=& -9 \\ \\ {- 3} && {- 3} \\ \\ x &=& -9 - 3 \end{eqnarray} $ $ x = -12 $ Then calculate the solution where $x + 3$ is positive: $ x + 3 = 9 $ Subtract ${3}$ from both sides: $ \begin{eqnarray} x + 3 &=& 9 \\ \\ {- 3} && {- 3} \\ \\ x &=& 9 - 3 \end{eqnarray} $ $ x = 6 $ Thus, the correct answer is $x = -12 $ or $x = 6 $.